An Introduction to the Theory of Higher-Dimensional Quasiconformal Mappings

The authors — Gehring [1], Martin, and Palka (GMP) — have written an excellent monograph for students aiming to learn the rudiments of quasiconformal (QC) maps, without an exclusive emphasis on the theory in two real dimensions. Potential readers of GMP’s text will be drawn to the minimal prerequisites. The book is accessible to readers familiar with real several-variable calculus, say at the level of Apostol’s Mathematical Analysis. In addition, the authors are careful to include a myriad details that are often swept under the rug, and that plenty of students will undoubtedly be grateful for.

The theory of QC maps was designed as an optimal flexibilization of the theory of conformal (angle-preserving) mappings in the plane. While conformal maps are characterized by the geometric property that infinitesimal balls are mapped to infinitesimal balls, QC maps transform infinitesimal balls to infinitesimal ellipsoids of bounded eccentricity. More precisely, a homeomorphism \(f : \mathbb{R}^n \to \mathbb{R}^n\) is called \(K\)-quasiconformal (for some \(K \geq 1\)) if for every \(x \in \mathbb{R}^n\) \[ \limsup_{r \to 0} \frac{\sup\{ |f(x) - f(y)| : |x- y| \leq r\}}{~\inf\{ |f(x) - f(y)| : |x- y| \geq r \}} \leq K. \] This is one path to characterizing QC maps though there are others (and proving equivalences is non-trivial!), e.g., via certain PDEs that extend the Cauchy-Riemann equations, or as homeomorphisms that change a conformal invariant (e.g., extremal length/conformal modulus) by a bounded amount. Though this last approach to defining QC maps is somewhat technical on the surface, it has several pedagogical advantages and the authors take this route to carefully guide readers towards proving Gehring’s rigidity theorem, one of the gems of the higher dimensional QC theory.

Gehring’s theorem is a vast generalization of Liouville’s rigidity theorem, which states that every sufficiently smooth [2] conformal self-map of \(\mathbb{R}^n\) with \(n \geq 3\) is a Möbius map, i.e., a composition of a finite number of reflections in \((n - 1)\)-spheres and hyperplanes. Liouville’s theorem implies that the only domains conformally equivalent to the unit (\(n \geq 3\))-ball are round balls or half-spaces; while the Riemann mapping theorem states that any simply-connected proper subdomain of the complex plane is conformally equivalent to the \(2\)-ball. Gehring’s extension proves that a homeomorphism \( f : \mathbb{R}^n \to \mathbb{R}^n \) with \(n\geq3\) is a Möbius map if and only if \[ \limsup_{r \to 0} \frac{\sup\{ |f(x) - f(y)| : |x- y| \leq r\}}{~\inf\{ |f(x) - f(y)| : |x- y| \geq r \}} = 1 . \] Note that the map is only assumed to be a homeomorphism! Higher regularity (e.g., if \( f \) were a diffeomorphism) would afford an straightforward reduction to Liouville’s result.

After reviewing the rudiments of topology and analysis in \(\mathbb{R}^n\) the authors move to a chapter on conformal maps that ends with a complete proof of Liouville’s theorem. The authors follow a proof of Liouville’s theorem due to Rolf Nevanlinna (“On differentiable mappings”, Analytic Functions, ed. by L. Ahlfors et al., Princeton University Press (1960), 3–9). Nevanlinna’s proof contained a gap repeated by various authors since, e.g. in Huff’s Conformal maps on Hilbert space (Bull.~AMS 82 (1976), 147–149). The original mistake was in the proof of GMP’s Lemma 3.8.4 in the case \(\beta=0\). Nevanlinna had assumed that this implies that \(\rho\) is constant [3], whereas it only implies that \(\rho\) is linear. GMP get around this lacuna by applying their Lemma 3.8.2, which implies that \(\rho\) must be constant if it is linear. The authors take the first seventy-five pages of their book to conclude their proof of Liouville’s theorem. The proof of Gehring’s theorem follows some two hundred plus pages later. In between, there is a wealth of material regarding geometric and analytic aspects of QC theory. The authors carefully prove equivalences between three characterizations of QC maps, and students will appreciate that they have not shied way from providing details.

After Gehring’s rigidity result, the authors move to a chapter on mapping problems (that includes a proof for the Schoenflies theorem in the quasiconformal category) followed by one on the Tukia-Väisälä extension theorem (Annals of Math.~115 (1982), 331-348). The latter was a landmark theorem asserting that every QC self-map of the boundary of \(n\)-dimensional hyperbolic space is the boundary correspondence for a QC self-map of the interior (i.e. of \(n\)-dimensional hyperbolic space). GMP do not prove this in full detail: there is an important approximation result due to Sullivan used as a black-box. However, the reader is guided to an adequate reference by Tukia-Väisälä that contains “a fairly detailed exposition of Sullivan’s theory”.

The book ends with a chapter proving a base case of Mostow’s rigidity theorem, viz. that any two closed hyperbolic manifolds of dimension \(n \geq 3\) with isomorphic fundamental groups must be isometric! Topology surprisingly determines geometry in such realms. Readers impatient to see what goes in to proving such a result may be directed to Bourdon’s elegant lecture notes Quasiconformal geometry and Mostow rigidity (which takes certain results for granted to keep the exposition under 15 pages). Note that the approach delineated by Bourdon uses an ergodic theory result due to Moore for its denouement, which is a technique avoided by GMP. Instead, the authors

take a fairly roundabout approach here so as to be able to clearly exhibit the remarkable interaction between quasiconformal theory, hyperbolic geometry, and modern aspects of geometric group theory. In particular we give a fairly comprehensive discussion of quasi-isometries and isomorphisms of hyperbolic groups.

Students reading GMP may be unaware of the extent of activity following Mostow’s discovery in the late 1960s, and those interested in such developments could peruse Bourdon’s survey Mostow type rigidity theorems.

Though QC theory provides plenty of riches in higher-dimensional Euclidean realms and beyond, an emphasis on the planar theory seems to have dominated the pedagogical landscape. Various applications of planar QC methods and Teichmüller theory are foundational to workers in low-dimensional topology and dynamics inspired by Sullivan’s celebrated dictionary between the theory of iteration of rational maps on the Riemann sphere and the theory of Kleinian groups (discrete subgroups of isometries of three-dimensional hyperbolic space); as well as by Thurston’s seminal work on his eponymous hyperbolization theorems and his topological characterization of rational maps.

Students wanting to learn planar QC mapping theory usually find their way to Ahlfors’ Lectures on quasiconformal mappings (Second edition, AMS, 2006) and Lehto and Virtanen’s Quasiconformal mappings in the plane (Second edition, Springer, 1973). Nowadays such students could also reach out to newer texts with fresher emphases, e.g., Hubbard’s unfurling 4-volume opus Teichmüller Theory and Applications to Geometry, Topology, and Dynamics (Matrix Editions, 2006– ), and Astala, Iwaniec, and Martin’s Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane (Princeton 2009).

The standard higher-dimensional QC text was, for a number of years, Väisälä’s Lectures on \(n\)-dimensional quasiconformal mappings (Springer LNM vol. 229, 1971). This was followed by a number of texts on the larger class of noninjective QC maps (a.k.a. quasiregular maps): Reshetnyak’s Space mappings with bounded distortion (AMS, 1989) and Stability theorems in geometry and analysis (Kluwer, 1994), followed by Vuorinen’s Conformal geometry and quasiregular mappings (Springer LNM vol. 1319, 1988) and Rickman’s Quasiregular mappings (Springer,1993). For a recent overview of the major higher-dimensional developments one could turn to Martin’s The theory of quasiconformal mappings in higher dimensions, I.

There are predictably a few minor infelicities that occur in GMP’s text, but none that detract in any significant way from the book’s merits. E.g., there is a strange numbering in place for the subsections and theorems at the start of Chapter 4. The authors mistakenly state in the third paragraph of the preface: “In higher dimensions few manifolds admit a conformal structure, yet D. Sullivan has shown that every topological manifold admits a quasiconformal structure, that is, a covering with quasiconformal local coordinate charts.” This is not quite true, since Sullivan’s result does not apply to manifolds of dimension 4. The authors do get this right later in the book, in particular when they discuss this seminal result but sadly do not provide a proof. In fact, Sullivan went on in later joint work with Donaldson to sharpen the result by proving the existence of topological \(4\)-manifolds that do not admit any quasiconformal structure, as well as of smooth \(4\)-manifolds which are homeomorphic but not quasiconformally equivalent. The \(4\)-dimensional QC story is very far from well-understood. It would be useful to have a follow-up text that leads students to the brave new worlds inhabited by Sullivan’s constellation of results.

GMP are to be commended for writing their monograph with a remarkable attention to detail. Some instructors and students may find the lack of exercises to be a drawback. Nevertheless, the book could very profitably be used to lead a learning seminar. Readers keen to engage with the material could easily attempt to prove any of the results as exercises.

I am confident that this book will very soon become a standard basic reference.

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Notes

[1] Fred Gehring (1925–2012) was one of the founders of QC theory. Readers curious about his interesting life may enjoy Martin’s Frederick William Gehring, Life and Mathematics.

[2] In this book the authors prove it for maps that are \(C^4\).

[3] This corresponds to “Suppose first \(\alpha=0\). In this case we have \( \lambda = \frac1\rho = \text{const.}\)” in Nevanlinna (op.~cit.), where the roles of \( \alpha \) and \( \beta \) are reversed.

Tushar Das is an Associate Professor of Mathematics at the University of Wisconsin–La Crosse.